Optimal. Leaf size=74 \[ -\frac{5 b}{a^3 \sqrt{a+b x}}-\frac{5 b}{3 a^2 (a+b x)^{3/2}}+\frac{5 b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{7/2}}-\frac{1}{a x (a+b x)^{3/2}} \]
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Rubi [A] time = 0.0242279, antiderivative size = 80, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {51, 63, 208} \[ -\frac{5 \sqrt{a+b x}}{a^3 x}+\frac{10}{3 a^2 x \sqrt{a+b x}}+\frac{5 b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{7/2}}+\frac{2}{3 a x (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x^2 (a+b x)^{5/2}} \, dx &=\frac{2}{3 a x (a+b x)^{3/2}}+\frac{5 \int \frac{1}{x^2 (a+b x)^{3/2}} \, dx}{3 a}\\ &=\frac{2}{3 a x (a+b x)^{3/2}}+\frac{10}{3 a^2 x \sqrt{a+b x}}+\frac{5 \int \frac{1}{x^2 \sqrt{a+b x}} \, dx}{a^2}\\ &=\frac{2}{3 a x (a+b x)^{3/2}}+\frac{10}{3 a^2 x \sqrt{a+b x}}-\frac{5 \sqrt{a+b x}}{a^3 x}-\frac{(5 b) \int \frac{1}{x \sqrt{a+b x}} \, dx}{2 a^3}\\ &=\frac{2}{3 a x (a+b x)^{3/2}}+\frac{10}{3 a^2 x \sqrt{a+b x}}-\frac{5 \sqrt{a+b x}}{a^3 x}-\frac{5 \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x}\right )}{a^3}\\ &=\frac{2}{3 a x (a+b x)^{3/2}}+\frac{10}{3 a^2 x \sqrt{a+b x}}-\frac{5 \sqrt{a+b x}}{a^3 x}+\frac{5 b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0065211, size = 33, normalized size = 0.45 \[ -\frac{2 b \, _2F_1\left (-\frac{3}{2},2;-\frac{1}{2};\frac{b x}{a}+1\right )}{3 a^2 (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 67, normalized size = 0.9 \begin{align*} 2\,b \left ( -{\frac{1}{{a}^{3}} \left ( 1/2\,{\frac{\sqrt{bx+a}}{bx}}-5/2\,{\frac{1}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) }-1/3\,{\frac{1}{{a}^{2} \left ( bx+a \right ) ^{3/2}}}-2\,{\frac{1}{\sqrt{bx+a}{a}^{3}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47718, size = 494, normalized size = 6.68 \begin{align*} \left [\frac{15 \,{\left (b^{3} x^{3} + 2 \, a b^{2} x^{2} + a^{2} b x\right )} \sqrt{a} \log \left (\frac{b x + 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) - 2 \,{\left (15 \, a b^{2} x^{2} + 20 \, a^{2} b x + 3 \, a^{3}\right )} \sqrt{b x + a}}{6 \,{\left (a^{4} b^{2} x^{3} + 2 \, a^{5} b x^{2} + a^{6} x\right )}}, -\frac{15 \,{\left (b^{3} x^{3} + 2 \, a b^{2} x^{2} + a^{2} b x\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) +{\left (15 \, a b^{2} x^{2} + 20 \, a^{2} b x + 3 \, a^{3}\right )} \sqrt{b x + a}}{3 \,{\left (a^{4} b^{2} x^{3} + 2 \, a^{5} b x^{2} + a^{6} x\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 7.26392, size = 818, normalized size = 11.05 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19638, size = 88, normalized size = 1.19 \begin{align*} -\frac{5 \, b \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3}} - \frac{2 \,{\left (6 \,{\left (b x + a\right )} b + a b\right )}}{3 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3}} - \frac{\sqrt{b x + a}}{a^{3} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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